On the influence of the geometry on skin effect in electromagnetism

TL;DR

This paper analyzes how the geometry of a domain influences the skin effect in electromagnetism, deriving asymptotic expansions and confirming them through numerical experiments, highlighting the role of boundary curvature.

Contribution

It introduces a multiscale asymptotic expansion for Maxwell equations in domains with dielectric and conductor regions, linking skin depth to boundary curvature.

Findings

Skin depth depends on the mean curvature of the conductor boundary.

Asymptotic expansions accurately describe electromagnetic behavior in high conductivity regimes.

Numerical experiments confirm theoretical predictions for smooth and nonsmooth interfaces.

Abstract

We consider the equations of electromagnetism set on a domain made of a dielectric and a conductor subdomain in a regime where the conductivity is large. Assuming smoothness for the dielectric--conductor interface, relying on recent works we prove that the solution of the Maxwell equations admits a multiscale asymptotic expansion with profile terms rapidly decaying inside the conductor. This skin effect is measured by introducing a skin depth function that turns out to depend on the mean curvature of the boundary of the conductor. We then confirm these asymptotic results by numerical experiments in various axisymmetric configurations. We also investigate numerically the case of a nonsmooth interface, namely a cylindrical conductor.